Fixed point of multivalued mapping in uniform spaces

Transcription

1 Proc. Indian Acad. Sci. (Math. Sci.) Vol. 113, No. 2, May 2003, pp Printed in India Fixed point of multivalued mapping in uniform spaces DURAN TÜRKOGLU and BRIAN FISHER Department of Mathematics, Faculty of Science and Arts, University of Kirikkale, Yahsihan Kirikkale, Turkey Department of Mathematical Sciences, Faculty of Computer Sciences and Engineering, De Montfort University, The Gateway, Leicester, LE1 9BH, England duran MS received 31 December 2001; revised 10 July 2002 Abstract. In this paper we prove some new fixed point theorems for multivalued mappings on orbitally complete uniform spaces. Keywords. Fixed point; multivalued mappings; orbitally complete; uniform space. 1. Introduction Let (X, U) be a uniform space. A family {d i : i I} of pseudometrics on X with indexing set I, is called an associated family for the uniformity U if the family β = {V(i,ε):i I;ε>0}, V(i,ε)={(x, y) : x,y X, d i (x,y)<ε} is a sub-base for the uniformity U. We may assume that β itself is a base by adjoining finite intersection of members of β, if necessary. The corresponding family of pseudometrics is called an associated family for U. An associated family for U will be denoted by p.for details the reader is referred to [1,3 8]. Let A be a nonempty subset of a uniform space X. Define (A) = sup {d i (x, y) : x,y A, i I}, {d i : i I} = p. Then is called an augmented diameter of A. Further, A is said to be p -bounded if (A) <. Let 2 X = { A : A is a nonempty, closed and p - bounded subset of X }. For any nonempty subsets A and B of X, define d i (x, A) = inf {d i (x, a) : a A}, i I 183

2 184 Duran Türkoglu and Brian Fisher { } H i (A, B) = max sup d i (a, B), sup d i (A, b) a A b B = sup { d i (x, A) d i (x, B) }. x X It is well-known that on 2 X, H i is a pseudometric, called the Hausdorff pseudometric induced by d i,i I. Let (X, U) be a uniform space with an augmented associated family p. p also induces a uniformity U on 2 X defined by the base β = { V (i, ε) : i I, ε > 0 }, V (i, ε) = { } (A, B) : A, B 2 X,H i (A,B)<ε. The space (2 X, U ) is a uniform space called the hyperspace of (X, U). DEFINITION1 The collection of all filters on a given set X is denoted by (X).An order relation is defined on (X) by the rule F 1 < F 2 iff F 1 F 2.IfF <F, then F is called a subfilter of F. DEFINITION2 Let (X, U) be a uniform space defined by {d i : i I} =p.iff :X 2 X is a multivalued mapping, then (i) x X is called a fixed point of F if x Fx; (ii) An orbit of F at a point x 0 X is a sequence {x n } given by O(F,x 0 ) ={x n :x n Fx n 1,n=1,2,...}; (iii) A uniform space X is called F -orbitally complete if every Cauchy filter which is a subfilter of an orbit of F at each x X converges to a point of X. DEFINITION3 Let (X, U) be a uniform space and let F : X X be a mapping. A single-valued mapping F is orbitally continuous if lim (T n i x) = u implies lim T(T n ix) = Tufor each x X. 2. Main results Theorem 1. Let (X, U) be an F -orbitally complete Hausdorff uniform space defined by {d i : i I} = p and (2 X, U ) a hyperspace and let F : X 2 X be a continuous mapping with Fx compact for each x in X. Assume that {H i (F x, Fy) r,d i (x, F x)d i (y, Fy) r 1,d i (y, Fy) r} + a i {d i (x,fy),d i (y, F x)} [b i d i (x, F x) + c i d i (x, y)]d i (y, Fy) r 1 (1) for all i I and x, y X, r 1 is an integer, a i, b i,c i are real numbers such that 0 <b i +c i <1, then F has a fixed point.

3 Multivalued mapping in uniform spaces 185 Proof. Let x 0 be an arbitrary point in X and consider the sequence {x n } defined by x 1 Fx 0,x 2 Fx 1,..., x n Fx n 1,... Let us suppose that d i (x n,fx n )>0for each i I and n = 0, 1, 2,... (Otherwise for some positive integer n, x n Fx n as desired.) Let U U be an arbitrary entourage. Since β is a base for U, there exists V(i, ε) β such that V(i, ε) U.Nowy d i (x 0,y)is continuous on the compact set Fx 0 and this implies that there exists x 1 Fx 0 such that d i (x 0,x 1 )=d i (x 0,Fx 0 ). Similarly, Fx 1 is compact so there exists x 2 Fx 1 such that d i (x 1,x 2 )=d i (x 1,Fx 1 ). Continuing, we obtain a sequence {x n } such that x n+1 Fx n and d i (x n,x n+1 )=d i (x n,fx n ). For x = x n 1, and y = x n by condition (1) we have {H i (F x n 1,Fx n ) r,d i (x n 1,Fx n 1 )d i (x n,fx n ) r 1,d i (x n,fx n ) r} +a i {d i (x n 1,Fx n ), d i (x n,fx n 1 )} [ b i d i (x n 1,Fx n 1 ) + c i d i (x n 1,x n ) ] d i (x n, Fx n ) r 1 or since d i (x n,fx n 1 )=0,x n Fx n 1. Hence we have {d i (x n,x n+1 ) r,d i (x n 1,x n )d i (x n,x n+1 ) r 1} [ b i d i (x n 1,x n )+ c i d i (x n 1,x n ) ] d i (x n, x n+1 ) r 1 and it follows that {d i (x n,x n+1 ) r,d i (x n 1,x n )d i (x n,x n+1 ) r 1} Since (b i + c i )d i (x n 1,x n )d i (x n, x n+1 ) r 1. d i (x n 1,x n )d i (x n,x n+1 ) r 1 (b i + c i )d i (x n 1,x n )d i (x n, x n+1 ) r 1 is not possible (as 0 <b i +c i <1), we have or d i (x n,x n+1 ) r (b i + c i )d i (x n 1,x n )d i (x n, x n+1 ) r 1 d i (x n,x n+1 ) r k i d i (x n 1,x n )d i (x n, x n+1 ) r 1, k i = b i + c i, 0<k i <1. Proceeding in this manner we get d i (x n,x n+1 ) k i d i (x n 1,x n ) k 2 i d i(x n 2,x n 1 ). ki n d i(x 0,x 1 ).

4 186 Duran Türkoglu and Brian Fisher Hence we obtain d i (x n,x m ) d i (x n,x n+1 )+d i (x n+1, x n+2 ) + +d i (x m 1,x m ) (k n i + k n+1 i + +k m 1 i )d i (x 0,x 1 ) ki n (1+k i+ +ki m n 1 )d i (x 0,x 1 ) kn i 1 k d i(x 0,x 1 ). Since lim n kn i = 0, it follows that there exists N(i,ε) such that d i (x n,x m )<εand hence (x n,x m ) U for all n, m N(i,ε). Therefore the sequence {x n } is a Cauchy sequence in the d i -uniformity on X. { Let S p = {x n : n p} for all positive integers p and let β be the filter basis Sp : p = 1, 2,... }. Then since {x n } is a d i -Cauchy sequence for each i I, it is easy to see that the filter basis β is a Cauchy filter in the uniform space (X, U). To see this we first note that the family {V(i,ε):i I}is a base for U as p = {d i : i I}. Now since {x n } is a d i -Cauchy sequence in X, there exists a positive integer p such that d i (x n,x m )<ε for m p, n p. This implies that S p S p V(i,ε). Thus given any U U, we can find an S p β such that S p S p U. Hence β is a Cauchy filter in (X, U). Since (X, U) is F -orbitally complete and Hausdorff space, S p z for some z X. Consequently F(S p ) Fz(follows from the continuity of F ). Also S p+1 F(S p ) = {Fx n : n p} for p = 1, 2,... It follows that z Fz. Hence z is a fixed point of F. This completes the proof. If we take r = 1 in Theorem 1, then we obtain the following theorem. Theorem 2. Let (X, U) be an F -orbitally complete Hausdorff uniform space defined by {d i : i I} = p and (2 X, U ) a hyperspace, let F : X 2 X be a continuous mapping and Fx compact for each x in X. Assume that {H i (F x, Fy), d i (x,fx),d i (y, Fy)} + a i { d i (x,fy),d i (y, F x)} b i d i (x, F x) + c i d i (x, y) (2) for all i I and x, y X, a i,b i,c i are real numbers such that 0 <b i +c i <1, then F has a fixed point. We denote that if F is a single valued mapping on X, then we can write d i (F x, Fy) = H i (F x, Fy), x, y X, i I. Thus we obtain the following theorem as a consequence of the Theorem 2. Theorem 3. Let (X, U) be a T -orbitally complete Hausdorff uniform space and let T : X X be a T -orbitally continuous mapping satisfying {d i (T x, T y), d i (x,tx),d i (y, T y)} + a i { d i (x,ty),d i (y, T x)} b i d i (x, T x) + c i d i (x, y) (3) for all x, y X, i I and a i,b i,c i are real numbers such that 0 <b i +c i <1. Then T has a fixed point and which is unique whenever a i >c i >0.

5 Multivalued mapping in uniform spaces 187 Proof. Define a mapping F of X into 2 X by putting Fx ={Tx}for all x in X. It follows that F satisfies the conditions of Theorem 2. Hence T has a fixed point. Now if a i >c i >0, we show that T has a unique fixed point. Assume that T has two fixed points z and w which are distinct. Since d i (z, T z) = 0 and d i (w, T w) = 0, then by the condition (2), or a i { d i (z,tw),d i (w, T z)} c i d i (z, w) a i d i (z, w) c i d i (z, w), d i (z, w) c i a i d i (z, w) which is impossible. Thus if a i >c i >0, then T has a unique fixed point in X. This completes the proof. We note that if a i = 1 in condition (3), then one gets the following result as a corollary. COROLLARY4 Let T be an orbitally cotinuous self-map of a T -orbitally complete uniform space (X, U) satisfying the condition {d i (T x, T y), d i (x,tx),d i (y, T y)} {d i (x,ty),d i (y, T x)} b i d i (x, T x) + c i d i (x, y), x, y X, i I and 0 <b i +c i <1. Then for each x X, the sequence {T n x} converges to a fixed point of T. Remark 1. If we replace the uniform space (X, U) in Theorem 3 and Corollary 4 by a metric space (i.e. a metrizable uniform space), then Theorem 1 and Corollary 1 of Dhage [2] will follow as special cases of our results. Acknowledgement This research was supported by the Scientific and Technical Research Council of Turkey, TBAG-1742 (1999). References [1] Acharya S P, Some results on fixed point in uniform space, Yokohama Math. J. XXII (1) (1974) [2] Dhage B C, Some results for the maps with a nonunique fixed point, Indian J. Pure Appl. Math. 16(3) (1985) [3] Kelley J L, General Topology (Van Nonstrand Company Inc.) (1955) [4] Mishra S N and Singh S N, Fixed point of multivalued mapping in uniform spaces, Bull. Cal. Math. Soc. 77 (1985) [5] Taraftar E, An approach to fixed point theorems on uniform spaces, Trans. Amer. Math. Soc. 77 (1985) [6] Taylor W W, Fixed point theorems for nonexpansive mappings in linear topological spaces, J. Math. Anal. Appl. 40 (1972) [7] Thron W J, Topological structures (New York: Holt, Rinehart and Winston) (1966) [8] Türkoglu D, Özer O and Fisher B, Some fixed point theorems for set valued mapping in uniform spaces, Demonstratio Math. 2 (1999)